Tatyana Sorokina

A multivariate Powell–Sabin interpolant

We consider the problem of constructing a C1 piecewise quadratic interpolant, Q, to positional and gradient data defined at the vertices of a tessellation of n-simplices in

C 1 Quintic Splines on Type-4 Tetrahedral Partitions.

\mathbb{R}^{n}

Smooth macro-elements on Powell-Sabin-12 splits

. The key to the interpolation scheme is to appropriately subdivide each simplex to ensure that certain necessary geometric constraints are satisfied by the subdivision points. We establish these constraints using the Bernstein–Bézier form for polynomials defined over simplices, and show how they can be satisfied. When constructed, the interpolant Q has full approximation power.

An explicit quasi-interpolation scheme based on C 1 quartic splines on type-1 triangulations

Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type-4) partition and associated trivariate C1 quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.

A C1 quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions

Macro-elements of smoothness C r are constructed on Powell-Sabin-12 splits of a triangle for all r ≥ 0. These new elements complement those recently constructed on Powell-Sabin-6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary-value problems and for interpolation of Hermite data.

Intrinsic supersmoothness of multivariate splines

We describe a new scheme based on quartic C^1-splines on type-1 triangulations approximating regularly distributed data. The quasi-interpolating splines are directly determined by setting the Bernstein-Bezier coefficients of the splines to appropriate combinations of the given data values. Each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Moreover, the operator interpolates the given data values at all the vertices of the underlying triangulation. Since the Bernstein-Bezier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield nearly-optimal approximation order for smooth functions. The order is known to be best possible for these spaces. Numerical tests confirm the theoretical behavior and show that the approach leads to functional surfaces of high visual quality.

Two tetrahedral C1 cubic macro elements

In 1988, Worsey and Piper constructed a trivariate macro-element based on C^1 quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macro-element spline space over tetrahedral partitions that satisfy some very restrictive geometric constraints. We show that by further refining their split, it is possible to construct a macro-element also based on C^1 quadratic splines that can be used with arbitrary tetrahedral partitions. The resulting macro-element space is stable and provides full approximation power.

Multivariate \(C^1\)-Continuous Splines on the Alfeld Split of a Simplex

We show that many spaces of multivariate splines possess additional smoothness (supersmoothness) at certain faces where polynomial pieces join together. This phenomenon affects the dimension and interpolating properties of splines spaces. The supersmoothness is caused by the geometry of the underlying partition.

Two condensed macro-elements with full approximation power

We propose two tetrahedral C^1 cubic macro elements that are constructed locally on one tetrahedron without any knowledge of the geometry of neighboring tetrahedra. Among such geometrically unconstrained local polynomial tetrahedral C^1 schemes requiring only first order derivative data, our macro elements have the smallest number of coefficients. The resulting macro element spaces are stable and provide full approximation power. We give explicit formulae that can be used to implement our schemes.

Full length article: Optimal recovery of twice differentiable functions based on symmetric splines

Using algebraic geometry methods and Bernstein-Bezier techniques, we find the dimension of \(C^1\)-continuous splines on the Alfeld split of a simplex in \(\mathbb {R}^n\) and describe a minimal determining set for this space.